The chiral critical locus and topological structures

We study a differential graded VOA associated to the derived critical locus of a function $f$ on a smooth oriented $D$-dimensional variety $(X,\mathbf{vol})$. Informally, this VOA, $\mathbf{crit}^{ch}_{f}$, is just the algebra of chiral differential operators on the derived critical locus $\mathbf{crit}_{f}$. We prove, using a generalization of a physical construction of Witten, the $\mathbf{crit}^{ch}_{f}$ admits a \emph{topological structure} if $f$ is homogeneous for a $\mathbf{G}_{m}$ action on $(X,\mathbf{vol})$. If $\mathbf{vol}$ has weight $b$ and $f$ has weight $a$, we compute the rank of the topological structure in terms of the discrete invariants of the theory to be $$d=\Big(D-\frac{2b}{a}\Big).$$ We conclude with some remarks about BV quantization and a simple computation of characters.